Nuclear magnetic resonance (NMR) well logs are commonly used to interpret rock pore size and provide essential information for petrophysical evaluation. For a porous rock of porosity φ and hydraulic permeability k, the time-domain spin echo trains M(t) measured by the NMR antenna is usually decomposed as a N-exponential series of partial porosities (amplitudes) {Aj}j=1 . . . N corresponding to relaxation times {T2j}j=1 . . . N scaled logarithmically:
                              M          ⁡                      (            t            )                          =                              ∑                          j              =              1                        N                    ⁢                                    A              j                        ⁢                                          exp                ⁡                                  (                                                            -                      t                                        /                                          T                                              2                        ⁢                        j                                                                              )                                            .                                                          (        1        )            
The formed distribution of {Aj}j=1 . . . N as a function of {T2j}j=1 . . . N is usually called the T2 distribution and is often mapped to the distribution of pore sizes encountered in the porous rock. Assuming the pore space is saturated with light liquids whose bulk relaxation times are in the order of or more than 1000 ms, the overall T2 relaxation of fluids in pores is dominated by the surface relaxation mechanism: larger values of T2 correspond to larger pores, and smaller values correspond to smaller pores.
The relative amplitude Aj assigned to each T2j bin corresponds to a fractional porosity assigned to a given pore size range. If a free-fluid relaxation cutoff is known, integration of the T2 distribution amplitudes for values of T2 under that cutoff provide the bound-water index (BVI) of the rock, while free-fluid index (FFI) is obtained above that cutoff:
                              BVI          =                                    ∑                              j                =                1                                            N                cutoff                                      ⁢                          A              j                                      ⁢                                  ⁢        and                            (        2        )                                FFI        =                              ∑                          j              =                              N                cutoff                                      N                    ⁢                      A            j                                              (        3        )            knowing that the two pore volumes add up to the total rock porosityBVI+FFI=φ.  (4)
Using a single T2 cutoff to calculate bound-water index is simple and practical. However fluid mobility is not just dependent on pore throat size, but also on the amount of wetting fluid in the pores. One introduces a saturation profile to more accurately represent bound fluid changes with pore size or T2 relaxation, and a commonly used form of saturation profile is the generalized logistic function of T2:
                              S          ⁡                      (                          T              2                        )                          =                                                            a                1                            -                              a                2                                                                    (                                  1                  +                                                            T                      2                                        /                                          T                                              2                        ⁢                        cutoff                                                                                            )                            α                                +                      a            2                                              (        5        )            where default values are for the parameters: a1=1, a2=0, α=2 and T2cutoff=33 ms. This saturation profile is characteristic of a given rock facies. The spectral BVI (SBVI) for a given saturated rock sample within that rock facies can be calculated from the T2 distribution of that sample as follows:
                    SBVI        =                                            ∑                              j                =                1                            N                        ⁢                                          A                j                            ⁢                              S                ⁡                                  (                                      T                                          2                      ⁢                      j                                                        )                                                              =                                    ∑                              j                =                1                            N                        ⁢                                                            A                  j                                ⁡                                  (                                                                                                              a                          1                                                -                                                  a                          2                                                                                                                      (                                                      1                            +                                                                                          T                                                                  2                                  ⁢                                  j                                                                                            /                                                              T                                                                  2                                  ⁢                                  cutoff                                                                                                                                              )                                                α                                                              +                                          a                      2                                                        )                                            .                                                          (        6        )            
A spectral FFI (SFFI) is also obtained as follows:
                    SFFI        =                              ϕ            -            SBVI                    =                                    ∑                              j                =                1                            N                        ⁢                                                            A                  j                                ⁡                                  [                                      1                    -                                          (                                                                                                                                  a                              1                                                        -                                                          a                              2                                                                                                                                          (                                                              1                                +                                                                                                      T                                                                          2                                      ⁢                                      j                                                                                                        /                                                                      T                                                                          2                                      ⁢                                      cutoff                                                                                                                                                                  )                                                        α                                                                          +                                                  a                          2                                                                    )                                                        ]                                            .                                                          (        7        )            
As such BVI (or SBVI) may be quantified along with the corresponding irreducible water saturation:Swir=BVI/φ  (8)or in spectral form:SwirSBVI/φ.  (9)
The calibration of the cutoff value
      T          2      ⁢      cutoff        =            {              T                  2          ⁢          j                    }              j      =              N        cutoff            is usually accomplished through time-consuming special core analysis protocol that is not always consistent from vendor to vendor:                After proper cleaning, a cylindrical core plug representative of the studied rock facies is fully saturated with brine and then measured in a laboratory NMR spectrometer. This produces a first T2 distribution at 100% water saturation (Sw=100%).        The core plug is subsequently desaturated with non-wetting fluid (air or oil) following different procedures—usually through spinning of the sample until the maximum centrifuge speed is reached, or through centrifugation or porous-plate displacement until a target capillary pressure is met. The final fluid saturation level of his process is Swir, and may or may not be the right value that represents reservoir conditions.        A new NMR measurement at Sw=Swir defines the NMR signal of bound water; the value of T2cutoff is then determined as the T2 value where the bound water volume determined with the Sw=100% experiment (BVI as determined from Eq. (2) or SBVI as determined from Eq. (7)) is equal to the water volume remaining in the Sw=Swir experiment.        